Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle ABC = 8x + 27$, and $ m \angle CBD = 7x - 27$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {8x + 27} + {7x - 27} = {90}$ Combine like terms: $ 15x + 0 = 90$ Add $0$ to both sides: $ 15x = 90$ Divide both sides by $15$ to find $x$ $ x = 6$ Substitute $6$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 7({6}) - 27$ Simplify: $ {m\angle CBD = 42 - 27}$ So ${m\angle CBD = 15}$.